Transfer purée to a medium bowl and mix in egg white, rice, and reserved beans; season with salt and pepper. Form mixture into 6 patties about 1/2-inch thick; cover and chill 1 hour (this helps bind patties so they stay intact while cooking).

Heat 2 tablespoons oil in a large skillet over medium heat. Working in two batches and adding remaining 2 tablespoons oil between batches, cook patties until browned and crisp, about 5 minutes per side. Serve on buns with desired toppings.